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Properties

Label 3024.2.k.l
Level $3024$
Weight $2$
Character orbit 3024.k
Analytic conductor $24.147$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3024,2,Mod(1889,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.1889");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} - 45 x^{12} + 306 x^{11} - 378 x^{10} + 1704 x^{9} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + \beta_{12} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + \beta_{12} q^{7} - \beta_{10} q^{11} + (\beta_{15} + \beta_{4}) q^{13} - \beta_{5} q^{17} - \beta_{4} q^{19} - \beta_{14} q^{23} + ( - \beta_{7} + \beta_{3} + 2) q^{25} + ( - \beta_{14} + \beta_{13} + \cdots + \beta_{8}) q^{29}+ \cdots + ( - \beta_{15} - \beta_{12} + \cdots + \beta_{4}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{7} + 36 q^{25} - 8 q^{37} - 20 q^{43} + 2 q^{49} - 44 q^{67} - 40 q^{79} + 16 q^{85} - 36 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} - 45 x^{12} + 306 x^{11} - 378 x^{10} + 1704 x^{9} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 96\!\cdots\!71 \nu^{15} + \cdots - 90\!\cdots\!32 ) / 76\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 63679977 \nu^{15} - 438014494 \nu^{14} + 1443496954 \nu^{13} - 3073658820 \nu^{12} + \cdots - 128641878336 ) / 24698853056 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!79 \nu^{15} + \cdots + 87\!\cdots\!04 ) / 19\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19\!\cdots\!21 \nu^{15} + \cdots - 91\!\cdots\!48 ) / 19\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\!\cdots\!99 \nu^{15} + \cdots + 98\!\cdots\!88 ) / 15\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10\!\cdots\!63 \nu^{15} + \cdots - 45\!\cdots\!60 ) / 76\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14\!\cdots\!07 \nu^{15} + \cdots - 86\!\cdots\!52 ) / 96\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 28\!\cdots\!65 \nu^{15} + \cdots - 26\!\cdots\!40 ) / 15\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 42\!\cdots\!89 \nu^{15} + \cdots - 17\!\cdots\!92 ) / 15\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 24\!\cdots\!29 \nu^{15} + \cdots + 15\!\cdots\!84 ) / 76\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 29\!\cdots\!01 \nu^{15} + \cdots + 15\!\cdots\!56 ) / 76\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 33\!\cdots\!13 \nu^{15} + \cdots + 15\!\cdots\!88 ) / 76\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 21499912175367 \nu^{15} + 118714615744902 \nu^{14} - 331040469619086 \nu^{13} + \cdots + 13\!\cdots\!68 ) / 434694100132496 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 12\!\cdots\!41 \nu^{15} + \cdots - 75\!\cdots\!96 ) / 15\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 73\!\cdots\!01 \nu^{15} + \cdots + 34\!\cdots\!00 ) / 76\!\cdots\!88 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -2\beta_{12} + 2\beta_{11} + 2\beta_{10} + 3\beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - 2\beta_{4} - \beta_{2} + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{14} - 3\beta_{13} + 8\beta_{9} - 24\beta_{2} + 12\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{15} - 4 \beta_{12} - 4 \beta_{11} + 11 \beta_{9} - 11 \beta_{8} + 2 \beta_{6} - 7 \beta_{5} + \cdots + 8 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 67\beta_{15} - 112\beta_{12} + 8\beta_{11} + 4\beta_{7} + 65\beta_{6} + 128\beta_{4} - 44\beta_{3} + 204 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 84 \beta_{15} - 128 \beta_{14} - 4 \beta_{13} - 370 \beta_{12} + 250 \beta_{11} + 130 \beta_{10} + \cdots + 636 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -500\beta_{14} - 187\beta_{13} + 176\beta_{10} + 796\beta_{9} + 796\beta_{8} ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1090 \beta_{15} - 1768 \beta_{14} - 266 \beta_{13} + 2966 \beta_{12} - 4558 \beta_{11} + 1374 \beta_{10} + \cdots - 8588 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 9979 \beta_{15} + 2712 \beta_{12} - 17760 \beta_{11} - 1356 \beta_{7} + 7817 \beta_{6} + 23184 \beta_{4} + \cdots - 28012 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 13880 \beta_{15} - 10236 \beta_{12} - 10236 \beta_{11} - 18053 \beta_{9} + 18053 \beta_{8} + \cdots - 23184 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 72212 \beta_{14} - 20739 \beta_{13} + 37216 \beta_{10} - 18608 \beta_{9} + 278408 \beta_{8} + \cdots - 216636 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 175310 \beta_{15} - 297016 \beta_{14} - 68102 \beta_{13} + 705502 \beta_{12} - 445702 \beta_{11} + \cdots - 1423612 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 409507\beta_{12} - 409507\beta_{11} - 60853\beta_{7} + 222851\beta_{3} - 1060651 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 2205436 \beta_{15} + 3762880 \beta_{14} + 909452 \beta_{13} + 5542138 \beta_{12} - 8818194 \beta_{11} + \cdots - 17998140 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 11084276 \beta_{14} + 2912299 \beta_{13} - 6229776 \beta_{10} - 44295104 \beta_{9} + \cdots - 33252828 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 27689690 \beta_{15} + 20590108 \beta_{12} + 20590108 \beta_{11} - 34586683 \beta_{9} + \cdots - 47409992 \beta_1 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1889.1
0.453016 + 0.121385i
0.453016 − 0.121385i
−0.651359 + 2.43091i
−0.651359 − 2.43091i
0.924776 − 0.247793i
0.924776 + 0.247793i
0.916156 − 3.41914i
0.916156 + 3.41914i
3.41914 + 0.916156i
3.41914 − 0.916156i
0.247793 + 0.924776i
0.247793 − 0.924776i
−2.43091 − 0.651359i
−2.43091 + 0.651359i
0.121385 − 0.453016i
0.121385 + 0.453016i
0 0 0 −4.29876 0 −0.832811 − 2.51126i 0 0 0
1889.2 0 0 0 −4.29876 0 −0.832811 + 2.51126i 0 0 0
1889.3 0 0 0 −2.85593 0 2.30652 − 1.29613i 0 0 0
1889.4 0 0 0 −2.85593 0 2.30652 + 1.29613i 0 0 0
1889.5 0 0 0 −1.22223 0 1.48321 − 2.19091i 0 0 0
1889.6 0 0 0 −1.22223 0 1.48321 + 2.19091i 0 0 0
1889.7 0 0 0 −0.933004 0 −2.45692 − 0.981597i 0 0 0
1889.8 0 0 0 −0.933004 0 −2.45692 + 0.981597i 0 0 0
1889.9 0 0 0 0.933004 0 −2.45692 − 0.981597i 0 0 0
1889.10 0 0 0 0.933004 0 −2.45692 + 0.981597i 0 0 0
1889.11 0 0 0 1.22223 0 1.48321 − 2.19091i 0 0 0
1889.12 0 0 0 1.22223 0 1.48321 + 2.19091i 0 0 0
1889.13 0 0 0 2.85593 0 2.30652 − 1.29613i 0 0 0
1889.14 0 0 0 2.85593 0 2.30652 + 1.29613i 0 0 0
1889.15 0 0 0 4.29876 0 −0.832811 − 2.51126i 0 0 0
1889.16 0 0 0 4.29876 0 −0.832811 + 2.51126i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1889.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.k.l 16
3.b odd 2 1 inner 3024.2.k.l 16
4.b odd 2 1 1512.2.k.b ✓ 16
7.b odd 2 1 inner 3024.2.k.l 16
12.b even 2 1 1512.2.k.b ✓ 16
21.c even 2 1 inner 3024.2.k.l 16
28.d even 2 1 1512.2.k.b ✓ 16
84.h odd 2 1 1512.2.k.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.k.b ✓ 16 4.b odd 2 1
1512.2.k.b ✓ 16 12.b even 2 1
1512.2.k.b ✓ 16 28.d even 2 1
1512.2.k.b ✓ 16 84.h odd 2 1
3024.2.k.l 16 1.a even 1 1 trivial
3024.2.k.l 16 3.b odd 2 1 inner
3024.2.k.l 16 7.b odd 2 1 inner
3024.2.k.l 16 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3024, [\chi])\):

\( T_{5}^{8} - 29T_{5}^{6} + 215T_{5}^{4} - 391T_{5}^{2} + 196 \) Copy content Toggle raw display
\( T_{11}^{8} + 33T_{11}^{6} + 380T_{11}^{4} + 1764T_{11}^{2} + 2704 \) Copy content Toggle raw display
\( T_{13}^{8} + 55T_{13}^{6} + 920T_{13}^{4} + 5036T_{13}^{2} + 7744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 29 T^{6} + \cdots + 196)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - T^{7} + 7 T^{5} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 33 T^{6} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 55 T^{6} + \cdots + 7744)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 49 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 40 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 85 T^{6} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 129 T^{6} + \cdots + 190096)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 154 T^{6} + \cdots + 64516)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} + \cdots + 196)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 226 T^{6} + \cdots + 8340544)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 5 T^{3} + \cdots - 554)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 98 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 282 T^{6} + \cdots + 200704)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 189 T^{6} + \cdots + 272484)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 252 T^{6} + \cdots + 1327104)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 11 T^{3} + \cdots - 56)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 401 T^{6} + \cdots + 26543104)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 467 T^{6} + \cdots + 33918976)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} + \cdots - 512)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 325 T^{6} + \cdots + 498436)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 391 T^{6} + \cdots + 7529536)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 239 T^{6} + \cdots + 529984)^{2} \) Copy content Toggle raw display
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